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International Journal of Electrical Engineering. ISSN 0974-2158 Volume 3, Number 3 (2010), pp. 271--288 © International Research Publication House http://www.irphouse.com

Use of Matlab Toolboxes for Performance Characteristics of SEIG under Varying Capacitance

and Load Conditions

Shelly Vadhera and K.S. Sandhu

Department of Electrical Engineering, National Institute of Technology, Kurukshetra, Haryana, India.

Abstract

In the present paper Matlab toolboxes are employed to estimate the steady state performance of self excited induction generator [SEIG]. Efforts are made to predict the performance of such generator with different values of excitation capacitance under any operating condition. Simulated results as obtained using direct search toolbox, genetic algorithm toolbox and optimization toolbox are compared with experimental results on a test machine. A close comparison proves the superiority of genetic algorithm [GA] toolbox in contrast to others. Keywords: Excitation Capacitance; Matlab Toolboxes; Resistive load; Self-excited induction generator.

List of symbols

0C Excitation Capacitance (µF)

F Per-unit frequency f Frequency in hertz (Hz)

bf Base frequency (Hz)

cI Capacitive current (A)

1I Stator current (A)

LI Load current (A)

mI Magnetizing current (A)

2'I Rotor current referred to stator (A)

lsL Stator leakage inductance (H)

mL Magnetizing inductance (H)

272 Shelly Vadhera and K.S. Sandhu

lrL' Rotor leakage inductance referred to stator (H)

n Rotor speed (rpm)

rn Synchronous speed corresponding to the generated terminal frequency (rpm)

0P Output power (W)

LR Per-phase load resistance (Ω)

sR Per-phase stator resistance (Ω)

rR' Per phase rotor resistance referred to stator (Ω) s Slip υ Per unit speed Vg Air gap voltage (V) ω Angular frequency of ac supply (rad./sec.)

bω Base angular frequency (rad./sec.)0C

mX Magnetizing reactance (Ω)

lsX Per-phase stator leakage reactance (Ω)

lrX ' Per-phase stator leakage reactance referred to stator (Ω)

cbX Reactance of the excitation capacitor at the base frequency (Ω)

Introduction In the past century, it has been seen that the consumption of non-renewable sources of energy has caused more environmental damage than any other human activity. Electricity generated from fossil fuels such as coal and crude oil has led to high concentrations of harmful gases in the atmosphere. This has in turn led to many problems being faced today such as ozone depletion and global warming. The growing awareness of these problems led to heightened research efforts for developing alternative sources of energy for generation of electricity. The most desirable source would be one that is non-pollutant, available in abundance, and renewable, and can be harnessed at an acceptable cost in both large-scale and small-scale systems. The most promising source satisfying all these requirements is wind, a natural energy source [1]. Wind has been known to man as a natural source of mechanical power for long. Before the invention of the steam engine, wind power had been used for centuries in sailing ships and then for pumping water and grinding grain. Only since the beginning of the 20th century it is being used to generate electric power [2]. Wind turbines convert wind energy into mechanical energy, which then needs to be converted into the electrical form using generators. In conventional generating stations, synchronous machines are used, while the variable-speed nature of wind energy necessitates a different strategy, wherein induction machines are used. Wind power generation, as it stands today, is dominated by induction generators. About 85% of the wind generators today are induction generators. An induction generator is a type of electrical generator that is mechanically and electrically similar to a polyphase induction motor [3]. Induction generators produce electrical power when their shaft is rotated faster than

Use of Matlab Toolboxes for Performance Characteristics 273

the synchronous frequency of the equivalent induction motor. It is being seen that three phase induction machines besides being commonly used as drives in the industry, have earned much favour as wind generators because of qualities such as ruggedness, reliability, and manufacturing simplicity [4]. They constitute the largest segment in the wind power industry today. Two types of three-phase induction machines are used: the squirrel cage type and the wound rotor (slip-ring) type. In the kilowatt range, the former is favoured, whereas in the megawatt range, the latter is favoured. Their principles of operation are basically the same; they differ only with respect to their application. Induction generators are not self-exciting, meaning they require a source of reactive power to produce a rotating magnetic flux. In case of self excited induction generators (SEIG) this flux is established by a capacitor bank connected to the machine terminals, whereas in case of grid connected induction generators it draws magnetizing current from the grid itself. In this paper Matlab toolboxes are employed to estimate the steady state performance of self excited induction generator. The performance characteristics of SEIG under varying values of terminal capacitance and load resistance have been discussed. An analytical method is developed using stator-referred circuit model of a SEIG normalized to the base frequency. Experimental results using a test machine with given set up in Appendix-3 are presented and compared with corresponding simulated results obtained from Matlab toolboxes. i.e. Genetic Algorithm toolbox employing GA solver; Direct search toolbox employing pattern search solver and Optimization toolbox employing unconstrained minimization problem. Although researchers [5-7] are using GA to sort out the problems related to SEIG, still the authors feels that this is the first attempt to use three Matlab toolboxes simultaneously for the analysis purpose of SEIG. Circuit model of the SEIG An equivalent circuit of the induction machine facilitates the analysis and computation of its performance. A per-phase circuit model is sufficient for balanced operation. In the capacitor self excited induction generator, the capacitor is required to provide the reactive power, and no external constraint (frequency and/or flux) is imposed on the system. An uncontrolled self excited induction generator shows considerable variation in its terminal voltage, saturation level, and output frequency under varying load conditions. For convenience, the parameters of the circuit model are defined in terms of a base frequency. Fig.1shows the per-phase, steady-state, stator-referred equivalent circuit of a self-excited induction generator connected to a resistive load. A capacitor of capacitance Co is connected to provide the reactive power.


Figure 1: Circuit model of a self excited induction generator at stator frequency. With reference to Fig.1, the following voltage equations can be written: glsst VILjIRV ++= 11 ω (1)

”Eq. (1)” is rewritten as glsst VIjFXIRV ++= 11 (2)

where F= bωω / and X ls = isb Lω ; Dividing ”Eq. (2)” by F yields

F

VIjXI

F

R

F

V gls

st ++= 11 (3)

The per-unit slip is

n

nns r−

= (4)

The slip given in ”Eq. (4)” may be expressed as

F

vFs

−= (5)

where υ = n br n/

The voltage equation for the rotor circuit shown in Fig.1 is

22 ''''

ILjIs

RV lr

rg ω+⎟

⎠⎞

⎜⎝⎛= (6)

in terms of the parameters F andυ ”Eq. (6)”, becomes

22 ''''

IjXIvF

R

F

Vlr

rg +−

= (7)


At the stator terminals, the following current balance equation holds:

1 0t

tL

VI j C V

Rω= +

In terms of the parameter F, this equation becomes

1 2

1 1

( / ) /t

cb L

VI

F jX F R F

⎡ ⎤= +⎢ ⎥−⎣ ⎦

(8)

where )/1( 0CwX bcb = is the reactance of the excitation capacitor at the base

frequency.

Figure 2: The stator referred circuit model of a SEIG normalized to base frequency. ”Eq. (3)”, ”Eq. (7)” and ”Eq. (8)” redefined the parameters and the node voltages of the equivalent circuit shown in Fig. 1, This stator referred equivalent circuit, as indicated in Fig.2 is mapped in terms of the base frequency and is used for predicting the performance of a self excited induction generator. In general, the speed and the load are given parameters. The frequency, the excitation capacitance, and the magnetizing reactance constitute the set of unknown variables even when a desired terminal voltage is required to be maintained. Steady state analysis The equivalent circuit shown in Fig.2 forms the basis for investigating the steady state performance of a self excited induction generator supplying a balanced load. There are two basic approaches, namely, the loop impedance method and the nodal admittance method, to defining the performance equations [8-10]. The choice of the method is influenced by the objective of the analysis. This paper deals with loop impedance method.


Loop impedance method Since there is no emf source, applying kirchhoff’s emf law around the loop SRPQ in the circuit of Fig.2 yields 0)( 1 =++ IZZZ RSPRQP (9)

For the stator current to exist under the self-excited state, the sum of the impedances must be zero, i.e., 0=++ RSPRQP ZZZ (10)

By equating the real and the imaginary parts independently to zero, we obtain two simultaneous non-linear equations:

0),,(

0),,(

2

1

==

mcb

mcb

XXFf

XXFf (11)

Two equations can yield only two unknown variables. The key unknown variable in determining the performance of an induction generator is the per-unit frequency F. The second unknown variable is chosen to be the X cb for a known value of X m

corresponding to rated voltage and this can be computed using ”Eq. (12)” ( obtained from the synchronous run test on induction machine.)

( )1

2

g

m

K VX

K

−= (12)

Where K1 and K2 depends on the design of the machine. The functions given by ”Eq. (11)” then assume the forms ( ) ( ) 0543

2

2

3

11, =++++= XAFAXAFAFAFXf CbCbCb

(13)

( ) ( ) ( ) 0543

2

212, =++++= XBFBXBFBXBFXf CbCbCbCb

(14)

Where the constants are defined as, ( )RXRXXA LlsLmls

2

1 2 +−=

υ×−= AA 12

( )( )'3 RRRXXA rSLlsm+++=

'4 RRRA rLS=

( )( ) υ×++−= RRXXA SLlsm5

XXXB lsmls

2

1 2 += ( )( )XXRRRB mlsrSL

++= '2

υ×−= BB 13

( ) υ×+−= XXRRB lsmLS4

( )RRRB SLr+−= '5


”Eq. (13)” and ”Eq. (14)” are solved using various optimization techniques in order to obtain the values of X cb and F. Once the values of X cb and F are obtained,

the following circuit relations is used to determine the generator performance.

( )QP PR

g tQP

Z ZV V

Z

+= (15)

rr

gm jXvFR

FVII

11 ')/('

/

+−+= (16)

L

tL R

VI = (17)

The output power is

LL RIP2

0 3= (18)

Thus for a known values of Vt, Vg, Xc, RL, υ and the generators equivalent circuit parameters the performance of SEIG can be computed. In this paper the performance of SEIG under varying values of load resistance and capacitances are predicted using various Matlab toolboxes. Matlab toolboxes In this paper three MATLAB toolboxes are selected because of their higher computational efficiency and also because they give the user a greater freedom in selecting the different operators. The Matlab Toolboxes selected are i.e. Genetic Algorithm toolbox employing Genetic Algorithm; Direct search toolbox employing pattern search tool and Optimization toolbox employing unconstrained minimization problem. The working of these toolboxes is given in the flowcharts in Appendix-2. Genetic algorithm toolbox One of the most important advantages of the Genetic Algorithm over the standard techniques is that it is able to find the global minimum, instead of a local minimum, and that the initial attempts with different starting point need not be close to actual values [11-12]. Another advantage is that it does not require the use of the derivative of the function, which is not always easily obtainable. Genetic Algorithm (GA) Toolbox searches for an unconstrained minimum of a function using the genetic algorithm. The GA works on a population using a set of operators that are applied to the population. A population is a set of points in the design space. The initial population is generated randomly by default. The next generation of the population is computed using the fitness of the individuals in the current generation. GA starts with a random set of points in the population and uses operators to produce the next generation of the population. The different operators are scaling, selection, crossover, and mutation. The best function value may improve or it may get worse by choosing different operators. Choosing a good set of operators for a


problem is often best done by experimentation. Thus GA tool provides a wonderful environment for easily experimenting with different options. The following outline summarizes how the GA toolbox works:

• The algorithm begins by creating a random initial population. • The algorithm then creates a sequence of new populations, or generations. At

each step, the algorithm uses the individuals in the current generation to create the next generation. To create the new generation, the algorithm performs the following steps:

o Scores each member of the current population by computing its fitness value.

o Scales the raw fitness scores to convert them into a more usable range of values.

o Selects parents based on their fitness. o Produces children from the parents. Children are produced either by

making random changes to a single parent called mutation or by combining the vector entries of a pair of parents called crossover.

o Replaces the current population with the children to form the next generation.

• The algorithm stops when one of the stopping criteria is met. Direct search toolbox Direct search is a method for solving optimization problems that does not require any information about the gradient of the objective function. As opposed to more traditional optimization methods that use information about the gradient or higher derivative to search for an optimal point, a direct search algorithm searches a set of points around the current point, looking for one where the value of the objective function is lower than the value at the current point. Direct search can be used to solve problems for which the objective function is not differential, or even continuous. Pattern Search is a subclass of direct search algorithms, which involve the direct comparison of objective function values and do not require the use of explicit or approximate derivatives. Pattern Search finds a linearly constrained minimum of a function. The Direct search Toolbox employing pattern search solver takes at least two input arguments, namely the objective function and a start point. The pattern search algorithm uses a set of rational basis vectors to generate search directions. It performs a search along the search directions using the current mesh size. The solver starts with an initial mesh size of one by default. A mesh can be scaled to improve the minimization of a badly scaled optimization problem. Scale is used to rotate the pattern by some degree and scale along the search directions. Direct search methods require many function evaluations as compared to derivative-based optimization methods. The pattern search algorithm can quickly find the neighborhood of an optimum point, but may be slow in detecting the minimum itself. This is the cost of not using derivatives. The Pattern Search solver can reduce the number of function evaluations using an accelerator. The following outline summarizes how the direct search toolbox works:


Meshes formation At each step, the pattern search algorithm searches a set of points, called a mesh, for a point that improves the objective function. The algorithm forms the mesh by:

1. Multiplying the pattern vectors by a scalar, called the mesh size. 2. Adding the resulting vectors to the current point (the point with the best

objective function value found at the previous step). Pattern Search polling At each step, the algorithm polls the points in the current mesh by computing their objective function values. By default the algorithm stops polling the mesh points as soon as it finds a point whose objective function value is less than that of the current point. The poll is then called successful and that point becomes the current point at the next iteration. After a successful poll, the algorithm multiplies the current mesh size by 2, the default value of mesh expansion factor. Because the initial mesh size is 1, at the second iteration the mesh size is 2. If the algorithm fails to find a point that improves the objective function, the poll is called unsuccessful and the current point stays the same at the next iteration. After an unsuccessful poll, the algorithm multiplies the current mesh size by 0.5, the default value of mesh contraction factor. The algorithm then polls with a smaller mesh size. Pattern Search Stopping Conditions The algorithm stops when any of the following conditions occurs:

1. The mesh size is less than mesh tolerance. 2. The number of iterations performed by the algorithm reaches the value of

maximum iteration. 3. The total number of objective function evaluations performed by the algorithm

reaches the value of maximum function evaluations. 4. The distance between the point found at one successful poll and the point

found at the next successful poll is less than X tolerance. 5. The change in the objective function from one successful poll to the next

successful poll is less than function tolerance. Optimization toolbox The Optimization Toolbox is a collection of functions that extend the capability of the Matlab numeric computing environment. The toolbox includes routines for many types of optimization including: Unconstrained nonlinear minimization, Constrained nonlinear minimization, including goal attainment problems, minimax problems, and semi-infinite minimization problems etc. Optimization Toolbox separates medium scale algorithms from large scale algorithms. The Optimization Toolbox offers a choice of algorithms and line search strategies. Under medium scale the principal algorithms used for unconstrained minimization are the Nelder-Mead simplex search method and the BFGS quasi-Newton method. Whereas for unconstrained minimization, minimax, goal attainment, semi-infinite optimization and variations of Sequential Quadratic Programming are used. The simplex search of Nelder and


Mead are most suitable for problems that are very nonlinear or have a number of discontinuities otherwise BFGS quasi-Newton method is preferred. Quasi-Newton methods are based on Newton's method to find the stationary point of a function, where the gradient is 0. Newton's method assumes that the function can be locally approximated as a quadratic in the region around the optimum, and use the first and second derivatives (gradient and Hessian) to find the stationary point. In quasi-Newton methods the Hessian matrix of second derivatives of the function to be minimized does not need to be computed at any stage. The Hessian is updated by analyzing successive gradient vectors instead. Quasi-Newton methods build up curvature information at each iteration to formulate a quadratic model problem of the form:

bxcHxx TTx ++

2

1min

where the Hessian matrix, H, is a positive definite symmetric matrix, c is a constant vector, and b is a constant. The optimal solution for this problem occurs when the partial derivatives of x go to zero, i.e., ( ) 0f x Hx c∇ = + = . The optimal solution

point, x can be written as cHx 1* −−= . Newton-type methods (as opposed to quasi-Newton methods) calculate H directly and proceed in a direction of descent to locate the minimum after a number of iterations. Calculating H numerically involves a large amount of computation. Quasi-Newton methods [13] avoid this by using the observed behavior of f(x) and ( )f x∇

to build up curvature information to make an approximation to H using an appropriate updating technique. A large number of Hessian updating methods have been developed. However, the formula of BFGS (Broyden, Fletcher, Goldfarb, and Shanno, given in 1970) is thought to be the most effective for use in a general purpose method. The formula given by BFGS is

1

T T TK K KK K K

K K T TK K KK K

q q H s s HH H

q s s H s+ = + −

Where 1K K Ks x x+= −

( ) ( )1K K Kq f x f x+= ∇ − ∇

As a starting point, 0H can be set to any symmetric positive definite matrix, for

example, the identity matrix I. At each iteration, k, a line search is performed in the direction, Kd where ( )1

k K Kd H f x−= − ∇ . Line search is a search method that is used as

part of a larger optimization algorithm. The method finds the next iterate of the form 1K K K Kx x d+ = + α . Here Kx denotes the current iterate, Kd is the search

direction, and Kα is a scalar step length parameter. At each iteration an update is


performed when a new point is found, 1Kx + , that satisfies the condition

( ) ( )1K Kf x f x+ < .

Results and discussions Fig.3 to Fig.10 shows the comparison of simulated results as obtained using direct search toolbox, genetic algorithm toolbox and optimization toolbox on a test machine (Appendix-1) that with results obtained from the experimental set up as shown in Appendix-4. Results obtained using genetic algorithm toolbox is found to be closer to experimental values. This indicates the superiority of GA in contrast to other techniques. Fig.3 to Fig.6 shows variation of terminal voltage with speed for various combinations of R L and X cb . It is observed that any variation in R L and X cb effects

the terminal voltage. Terminal voltage falls with load (i.e. R L decreases) and excitation capacitance.

Figure 3: Variation of terminal voltage with speed for values LR =3.5p.u. ,

cbX =2.0p.u.


cbX =2.0p.u.



cbX =1.0p.u.


cbX =1.0p.u.

Fig7-Fig.10 shows variation of frequency with speed for various combinations of R L and X cb . Decrease in frequency with decrease in the values of R L and vice versa

is proved from Fig.7 and Fig.8 and are further ensured by Fig.9 and Fig.10. Henceforth it can be concluded that there is fall in terminal voltage and frequency with corresponding changes in load resistance and capacitance.

Figure 7: Variation of frequency with speed for values LR =3.5p.u. , cbX =2.0p.u.





The simulated results as obtained using GA toolbox on a test machine are presented in Fig.11-Fig.14. Fig.11 shows the variation of terminal voltage, efficiency and frequency with output power at rated speed (υ =1pu.) and at a fixed value of


capacitance (X cb =1pu.). It may be noted that the terminal voltage and frequency falls

with output power whereas generator efficiency improves with load. Fig.12- Fig.14 shows the variation of terminal voltage, efficiency and frequency with output power for different values of excitation capacitance and machine operated at constant speed. Fig.12 shows fall in voltage with increase in load. This nature of the voltage is attributed to the armature reaction effect, impedance drop and the dependent excitation. Also it can be seen from the figure that the maximum power and terminal voltage rises with decrease in capacitive reactance X cb (or with increase in

capacitance values as X cb α 0

1

C) . Fig.13 shows that the maximum power and

efficiency increase in value with increase in X cb .Whereas Fig.14 shows that the

maximum power and frequency increases with decrease in X cb values.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Ter

min

al V

olta

ge, E

ffic

ienc

y an

d Fre

quen

cy (p.

u.)

Output power (p.u.)

Terminal Voltage

Frequency Efficiency

Figure 11: Variation of terminal voltage, efficiency and frequency with output power.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

Ter

min

al v

olta

ge (p.

u.)

Output power (p.u.)

Xcb = 2.0pu.

Xcb = 1.5pu. Xcb = 1.0pu.

Figure 12: Variation of terminal voltage with output power for different values of

cbX .


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.4

0.5

0.6

0.7

0.8

0.9

1

Eff

cien

cy (

p.u.

)

Output power (p.u.)

Xcb = 2.0pu.


Figure 13: Variation of efficiency with output power for different values of cbX .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

Fre

quen

cy (p.

u.)

Output power (p.u.)

Xcb = 2.0pu.


Figure 14: Variation of frequency with output power for different values of cbX .

Conclusion Self excited induction generator generally preferred to harness the wind energy, requires pre estimation of steady state performance under specific operating conditions. In this paper an attempt has been made to search for an appropriate MATLAB toolbox to predict the behavior of such generator. Comparison of simulated results using direct search toolbox, genetic algorithm toolbox and optimization toolbox with experimental results on a test machine proves the justification of GA toolbox for such analysis. It is observed that simulated results as obtained using the GA toolbox are found to be in closer agreement with the experimental results. Such studies may be useful to enhance the application of genetic algorithm toolbox for the solution of specific problems related to electrical machines. Appendix-1 Rating of Machine 2.2KW/3HP 3-phase, 230 Volts, 8.6 Amp,


4 pole, 50 Hz delta connected cage induction motor Base Quantities: Voltage/phase – 230 Volts Currents/phase – 4.96 Amp Impedance/phase – 46.37ohms Power – 1140.8 Watts Frequency - 50 Hz Speed – 1500 r. p. m. Equivalent Circuit Parameters RS = 0.0722 p.u. RR = 0.038 p.u. XS = XR = XL = 0.1046 p.u. XM (unsaturated) = 2.33 p.u. K1 = 1.6275 K2 = 0.3418 Appendix-2 Flowcharts for the Genetic Algorithm toolbox, Direct search toolbox and Optimization toolbox respectively.


Appendix-3 Experimental set up

References

[1] Bolton, H.R., and Nicodemou V.C., 1979, “Operation of self excited generators for windmill applications,’’ Proc. IEE., 126(9), pp. 815-820.

[2] Murthy, S.S., Singh, B.P., Nagamani, C., and Satyanarayana, K.V.V., 1988, “Studies on the use of conventional induction motor as self-excited induction generators,” IEEE Trans. EC., 3(4), pp. 842-848.

[3] Tandon, A.K., Murthy, S.S., and Berg, G.J., 1984, “Steady state analysis of capacitor self-excited induction generator,” IEEE Trans. on PAS., 103(3),pp. 612-618.

[4] Anagreh, Y.N. and Al-Kofahi, I.S., 2006, “Genetic algorithm- Based performance analysis of self excited induction generators,” IJ of Modelling and Simulation., 26(2), pp. 175-179.

[5] Dheeraj Joshi, Sandhu, K.S. and Soni M.K., 2009, “Voltage control of self-excited induction generator using genetic algorithm,” Turk J Elec & Comp Sci., 17(1), pp.87-97.

[6] Dheeraj Joshi, Sandhu, K.S. and Soni M.K., 2006, “Constant voltage constant frequency operation for a self-excited induction generator,” IEEE Trans. EC., 21 (1), pp. 228-234.

[7] Shridhar, L., Bhim Singh and Jha C.S., 1993, “A step towards improvements in the characteristics of self excited induction generator,” IEEE Trans. EC., 8(1), pp.40-46.

[8] Singh, S.P., Bhim Singh and Jain, M.P., 1990, “Performance characteristics and optimum utilization of a cage machine as capacitor excited induction generator,” IEEE Trans. EC., 5(4), 679-684.

[9] Murthy, S.S., Malik, O.P. and Tandon. A.K., 1982, “Analysis of self-excited induction generators,” Proc. IEE., pt.C, 129(6), pp. 260-265.


[10] Sette, S. and Boullart L., 2001, “Genetic programming: principles and applications,” Engineering Applications of Artificial Intelligence., 14, pp. 727-736.

[11] Shanno, D.F., 1970, “Conditioning of quasi-newton methods for function minimization,” Mathematics of Computing., 24, pp. 647-656.

[12] Bhadra, S.N., Kastha, D., and Banerjee, S., 2005, Wind electrical systems, New Delhi: Oxford University Press.

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